CS 188: Artificial Intelligence
Spring 2007
Lecture 14: Bayes Nets III
3/1/2007
Srini Narayanan – ICSI and UC Berkeley
Announcements
 Office Hours this week will be on Friday
 (11-1).
 Assignment 2 grading
 Midterm 3/13
 Review 3/8 (next Thursday)
 Midterm review materials up over the
weekend.
 Extended office hours next week
 (Thursday 11-1, Friday 2:30-4:30)
Representing Knowledge
Inference
 Inference: calculating some
statistic from a joint probability
distribution
 Examples:
L
R
B
 Posterior probability:
D
T
 Most likely explanation:
T’
Inference in Graphical Models
 Queries
 Value of information
 What evidence should I seek next
 Sensitivity Analysis
 What probability values are most critical
 Explanation
 e.g., Why do I need a new starter motor
 Prediction
 e.g., What would happen if my fuel pump stops
working
Inference Techniques
 Exact Inference
 Inference by enumeration
 Variable elimination
 Approximate Inference/ Monte Carlo




Prior Sampling
Rejection Sampling
Likelihood weighting
Monte Carlo Markov Chain (MCMC)
Reminder: Alarm Network
Inference by Enumeration
 Given unlimited time, inference in BNs is easy
 Recipe:
 State the marginal probabilities you need
 Figure out ALL the atomic probabilities you need
 Calculate and combine them
 Example:
Example
Where did we
use the BN
structure?
We didn’t!
Example
 In this simple method, we only need the
BN to synthesize the joint entries
Nesting Sums
 Atomic inference is extremely slow!
 Slightly clever way to save work:
 Move the sums as far right as possible
 Example:
Evaluation Tree
 View the nested sums as a computation tree:
 Still repeated work: calculate P(m | a) P(j | a) twice, etc.
Variable Elimination: Idea
 Lots of redundant work in the computation tree
 We can save time if we carry out the summation
right to left and cache all intermediate results
into objects called factors
 This is the basic idea behind variable elimination
Basic Objects
 Track objects called factors
 Initial factors are local CPTs
 During elimination, create new factors
 Anatomy of a factor:
Variables
introduced
Variables
summed out
Factor
argument
variables
Basic Operations
 First basic operation: join factors
 Combining two factors:
 Just like a database join
 Build a factor over the union of the domains
 Example:
Basic Operations
 Second basic operation: marginalization
 Take a factor and sum out a variable
 Shrinks a factor to a smaller one
 A projection operation
 Example:
Example
Example
General Variable Elimination
 Query:
 Start with initial factors:
 Local CPTs (but instantiated by evidence)
 While there are still hidden variables (not Q or evidence):
 Pick a hidden variable H
 Join all factors mentioning H
 Project out H
 Join all remaining factors and normalize
Variable Elimination
 What you need to know:
 VE caches intermediate computations
 Polynomial time for tree-structured graphs!
 Saves time by marginalizing variables as soon as
possible rather than at the end
 Approximations
 Exact inference is slow, especially when you have a
lot of hidden nodes
 Approximate methods give you a (close) answer,
faster
Sampling
 Basic idea:
 Draw N samples from a sampling distribution S
 Compute an approximate posterior probability
 Show this converges to the true probability P
 Outline:
 Sampling from an empty network
 Rejection sampling: reject samples disagreeing with evidence
 Likelihood weighting: use evidence to weight samples
Prior Sampling
Cloudy
Sprinkler
Rain
WetGrass
Prior Sampling
 This process generates samples with probability
…i.e. the BN’s joint probability
 Let the number of samples of an event be
 Then
 I.e., the sampling procedure is consistent
Example
 We’ll get a bunch of samples from the BN:
c, s, r, w
c, s, r, w
c, s, r, w
c, s, r, w
c, s, r, w
Cloudy
C
Sprinkler
S
Rain
R
WetGrass
W
 If we want to know P(W)





We have counts <w:4, w:1>
Normalize to get P(W) = <w:0.8, w:0.2>
This will get closer to the true distribution with more samples
Can estimate anything else, too
What about P(C| r)? P(C| r, w)?
Rejection Sampling
 Let’s say we want P(C)
 No point keeping all samples around
 Just tally counts of C outcomes
 Let’s say we want P(C| s)
 Same thing: tally C outcomes, but
ignore (reject) samples which don’t
have S=s
 This is rejection sampling
 It is also consistent (correct in the
limit)
Cloudy
C
Sprinkler
S
Rain
R
WetGrass
W
c, s, r, w
c, s, r, w
c, s, r, w
c, s, r, w
c, s, r, w
Likelihood Weighting
 Problem with rejection sampling:
 If evidence is unlikely, you reject a lot of samples
 You don’t exploit your evidence as you sample
 Consider P(B|a)
Burglary
Alarm
 Idea: fix evidence variables and sample the rest
Burglary
Alarm
 Problem: sample distribution not consistent!
 Solution: weight by probability of evidence given parents
Likelihood Sampling
Cloudy
Sprinkler
Rain
WetGrass
Likelihood Weighting
 Sampling distribution if z sampled and e fixed evidence
Cloudy
C
 Now, samples have weights
S
Rain
R
W
 Together, weighted sampling distribution is consistent
=
Likelihood Weighting
 Note that likelihood weighting
doesn’t solve all our problems
 Rare evidence is taken into account
for downstream variables, but not
upstream ones
 A better solution is Markov-chain
Monte Carlo (MCMC), more
advanced
 We’ll return to sampling for robot
localization and tracking in dynamic
BNs
Cloudy
C
S
Rain
R
W
Summary
 Exact inference in graphical models is
tractable only for polytrees.
 Variable elimination caches intermediate
results to make the exact inference process
more efficient for a given for a given query.
 Approximate inference techniques use a
variety of sampling methods and can scale to
large models.
 NEXT: Adding dynamics and change to
graphical models
Bayes Net for insurance